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2012 AMC 10B Problem 6

Problem 6 of 25EasierAlgebra

In order to estimate the value of x−yx-y where xx and yy are real numbers with x>y>0,x > y > 0, Xiaoli rounded xx up by a small amount, rounded yy down by the same amount, and then subtracted her rounded values. Which of the following statements is necessarily correct?

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Solution

Let the small rounding amount be d>0.d>0. Xiaoli replaces xx by x+dx+d and yy by y−d.y-d. Her estimate is therefore (x+d)−(y−d)=x−y+2d,(x+d)-(y-d)=x-y+2d, which is larger than x−y.x-y. Thus, the correct answer is A .

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Concepts: estimation · inequality

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.